Relational Contracts with Private Information
On the Future Value of the Relationship
The Upside of Implicit Downsizing Costs
Matthias Fahn (JKU Linz)
Nicolas Klein (University of Montreal)
Discussion Paper No. 106
July 23, 2018
Collaborative Research Center Transregio 190 | www.rationality-and-competition.de Ludwig-Maximilians-Universität München | Humboldt-Universität zu Berlin
Relational Contracts with Private Information
on the Future Value of the Relationship:
The Upside of Implicit Downsizing Costs
∗
Matthias Fahn
†and Nicolas Klein
‡July 19, 2018
Abstract
We analyze a relational contracting problem, in which the principal has private information about the future value of the relationship. In order to reduce bonus payments, the principal is tempted to claim that the value of the future relationship is lower than it actually is. To induce truth-telling, the optimal relational contract may introduce distortions after a bad report. For some levels of the discount factor, output is reduced by more than would be sequentially optimal. This distortion is attenuated over time even if prospects remain bad. Our model thus provides an alternative explanation for indirect short-run costs of downsizing.
∗We thank Daniel Barron, Catherine Bobtcheff, Sylvain Chassang, Florian Englmaier,
Willie Fuchs, Tahmina Hadjer, Marina Halac, Martin Hellwig, Richard Holden, Johannes H¨orner, Sebastian Kranz, Nicolas Lambert, Steve Leider, Jin Li, Thomas Mariotti, Niko Matouschek, Aniko ¨Ory, Mike Powell, Sven Rady, Markus Reisinger, Andreas Roider, Larry Samuelson, Klaus Schmidt, Marco Schwarz, Robert Ulbricht, as well as attendees at sem-inars at the Universtitat de Barcelona, the MPI Bonn, the ETH Zurich, McGill, Guelph, the Universit´e Laval, the Universit´e de Montr´eal, Munich, Paris (S´eminaire Roy), Regens-burg, Toulouse, the UNSW, and at various conferences, for helpful comments. Matthias Fahn gratefully acknowledges financial support from the German Science Foundation (DFG) through collaborative research center CRC TRR 190. Nicolas Klein gratefully acknowledges financial assistance from the Fonds de Recherche du Qu´ebec Soci´et´e et Culture, and the Social Sciences and Humanities Research Council of Canada.
†JKU Linz and CESifo, [email protected].
1
Introduction
In many instances, organizations face difficulties in providing the proper incen-tives to their members because performance cannot be verified, i.e., enforced by a court. As noted by the literature on relational contracts, however, the mutual dependence that repeated interaction between the same parties fosters may allow contracting parties to overcome these difficulties. This engenders an implicit, or “relational,” contract between them, whereby the principal “voluntarily” rewards the agent for his effort. As the worst the agent can do to the principal is to break off the relationship entirely, the most the principal can credibly promise as a reward is the value of the entire future relationship to her.
Our goal here is to analyze the workings of such relational contracts when, at the time of deciding on rewards, the principal knows more about the future development, and hence the value, of the relationship. Indeed, management may e.g. be better informed about the likely evolution of demand for a firm’s product than workers. In such a context, workers must trust management not to use its informational advantage against them, e.g. by fraudulently claiming a threat of future demand contraction to cut their bonus payments or even let go of them.
We show that an optimal relational contract in such a setting can lead to a dynamic that has been discussed in the strategic management literature, which has noted that downsizing often seems less effective than originally anticipated.1 The prevailing explanation for these implicit downsizing costs
seems to be that surviving employees tend to consider downsizing as a breach of a “psychological contract” (Love and Kraatz (2009)), and thus switch to a kind of punishment mode in response. As Cascio (1993) writes: “Study after study shows that following a downsizing, surviving employees become narrow-minded, self-absorbed, and risk averse. Morale sinks, productivity drops, and survivors distrust management.” Love and Kraatz (2009) write: “Though downsizing was perfectly legal and widely advocated as an efficient business practice, it connoted opportunism and signaled that a firm was an untrustworthy actor that might not be counted on to meet its commitments in the future. Employees clearly interpreted downsizing as a betrayal and characterized downsizers as untrustworthy.”
Yet, there is some evidence to suggest that this “punishment mode” only lasts for a limited period of time. Indeed, Goesaert, Heinz, and Vanormelin-gen (2015) show that firm performance tends to drop at the downsizing event, recovering at best to pre-downsizing levels afterwards. Meuse, Bergmann, Vanderheiden, and Roraff (2004) and Meuse and Dai (2013) also demonstrate that, while downsizing firms perform significantly worse than other firms by several financial measures, this difference gradually vanishes, eventually be-coming insignificant. Conducting a survey of employees of a large high-tech firm, Amabile and Conti (1999) find that productivity significantly declined during and immediately after the downsizing process, recovering again after a while. The survey paper by Datta, Guthrie, Basuil, and Pandey (2010) quotes several studies showing that the benefits of downsizing, if any, will materialize only 2-3 years after the downsizing event.
Our paper provides an alternative explanation for the temporary lack of effectiveness of downsizing. In this view, implicit downsizing costs do not indicate a lack of trustworthiness, nor do they result from punishment for a broken promise. Instead, they will arise as part of the path of play in an optimal relational contract, acting as a commitment device only to downsize when it is necessary to do so. As a consequence, these implicit downsizing costs allow for increased productivity in good times.
More specifically, our model starts from the standard relational-contracting framework, in which a principal and an agent interact repeatedly over time. The agent has to exert effort to produce output, which translates into a profit for the principal. Effort is costly to the agent. By assuming that the agent is risk neutral and effort costs are linear in the level of effort exerted, we can interpret our agent as representing the firm’s total workforce, which is made up of homogeneous workers.2 Only one-period contracts are possible;
these cannot condition on the agent’s effort choice, which, although observ-able, contains subjective aspects and is hence not contractible. As effort is perfectly observable by both parties to the relationship, however, continuation play can depend on the level of effort observed. In particular, the principal can pay the agent a discretionary bonus for choosing the right level of effort; this bonus can be enforced by the agent’s threat to leave the relationship if a
2This interpretation presupposes multilateral relational contracts, by which a deviation
in the relationship with one agent is punished by a complete loss of trust in all other relationships, see Levin (2002).
bonus payment that was due to him was reneged on. Therefore, the principal can only credibly commit to a bonus that is at most as high as the expected value of the continuation of the relationship to her. The principal’s profits, which are generated by the (publicly observable) output in a given period, depend on the binary state of the world (“type”) in that period, which is only privately observed by the principal. The effort level the principal wants to induce may thus well depend on the current state of the world. The type of the next period is privately observed by the principal before she decides on the current period’s bonus. She thus has some private information on the value of the continuation of the relationship when she decides whether to pay out the bonus, or to renege, and thus to end the relationship.
Our analysis shows that, even though there is only one-sided private in-formation, some surplus may optimally be destroyed along the path of play, leading to implicit downsizing costs. The goal of this arrangement is to deter the principal from mulcting the agent of the bonus due to him by understating the value of the continuation of the relationship. Indeed, lest the principal be tempted by such a deviation, continuation play following a pessimistic an-nouncement must be rendered sufficiently unattractive. One way of achieving this goal would be to force the principal to pay the agent a transfer whenever the continuation value is low. This, however, turns out not to be optimal in our setting, the reason being that this penalty would hurt a truthful on-path principal and a lying off-path principal alike. By contrast, a distortion in the agent’s effort hits an off-path principal, who has falsely claimed that effort is less productive, more than an on-path principal, who has been honest in invoking a low productivity of effort. Such an effort distortion reduces output and profits below levels that would be feasible at this point in time – which however is optimal because it allows to sustain higher output and profits in earlier, high-state, periods.
In a next step, we explore in Section 5 how the precise timing of informa-tion revelainforma-tion affects our outcomes. First, we assume that the state of the world is revealed later than in our main model, at the beginning of each re-spective period, i.e., after the previous period’s bonus payment is sunk. In this case, private information is not costly, and the principal can credibly promise the full expected continuation value as a means of providing incentives. This implies that it is feasible and optimal to make the agent’s compensation in a period independent of next period’s type. In a second step, we explore the
effects of the type being revealed earlier than in our main setting, assuming that next period’s state of the world is observed by the principal already at the beginning of the current period, before the agent has chosen current-period ef-fort. Now, the principal’s private information is again costly and truth-telling conditions constrain profits, albeit due to an issue that has been absent be-fore: Because a high state of the world in the next period potentially allows for higher effort – and consequently higher profits – in the current period, the principal has an incentive to misrepresent tomorrow’s type as high, and sub-sequently to renege on the promised payment. It turns out that, on account of this constraint, it is not possible to generate higher profits if tomorrow’s type is high. Those will only be a function of today’s type, and will always be constrained by the continuation value that would prevail if tomorrow’s type was low, irrespectively of whether tomorrow’s type ends up being high or low. In contrast to before, truth-telling can now also be achieved via fixed pay-ments made to the agent at the beginning of a period. Put differently, either effort will be independent of tomorrow’s state of the world, or a high state tomorrow also triggers higher effort, albeit with a fixed payment made to the agent before effort is delivered. This payment has to fully make up for the increased value of production.
Thus, when the principal strives to motivate the agent to exert effort, she is tempted to claim that the future looks bright and that hence the agent will be compensated for his hard work. Yet when the principal is supposed to compensate the agent, she is tempted to claim that the future looks grim – and that the agent will consequently have to accept lower compensation.
In most of the paper, we focus on the case in which the principal’s type is iid across periods. In this case, only a distortion in the next period hits an off-path principal more severely than an on-path principal. Consequently, im-plicit downsizing costs will only last for one period in this setting, after which effort increases to its undistorted level even if the firm’s prospects remain bad. Indeed, the management literature has noted that, at the occurrence of downsizing events, there is often some overshooting in the reduction of labor input, as evidenced by the fact that firms tend to increase labor input again shortly after downsizing, while the firm’s environment has not changed.3
In Section 6, we extend our analysis to (fully or partially) persistent
shocks. In these cases, distortions gradually attenuate over time but only ever vanish in the limit. The reason is that, with persistent shocks, an off-path principal is hit more severely by a distortion in any future period, but the difference in on-path vs. off-path costs diminishes with distance in time.
The idea that repeated interaction endogenously creates some scope for commitment via implicit contracts has been applied to labor markets by Bull (1987), as well as MacLeod and Malcomson (1989).4 These early papers
ab-stracted from informational asymmetries, focusing instead on the question of how incentives can be governed by non-contractual agreements. Levin (2003) augmented the analysis by introducing informational asymmetries, analyzing the cases in which the employee privately knows his effort costs (adverse se-lection), his effort level can only be imperfectly observed (moral hazard), as well as the case in which the employer privately observes a performance mea-sure, while not observing the agent’s effort choice directly. Malcomson (2016) introduces persistent types into Levin’s (2003) adverse-selection model, and finds that a full separation of types is not possible when continuation pay-offs are on the Pareto frontier. Malcomson (2015) augments Levin’s (2003) adverse-selection model by the introduction of different principal-types de-noting the productivity of the agent’s effort in the current period. At the time the principal decides on her bonus payment, however, she does not have any additional information concerning future productivity, in contrast to our setting. Halac (2012) analyzes the case of a principal who privately knows the value of her outside option while not being able to observe the agent’s effort level directly. In Halac (2012), there is no direct productive distortion in the agent’s not knowing the principal’s private information; in our setting, by contrast, the first-best level of effort depends on its productivity. In Li and Matouschek (2013), the principal has one-sided private information as well. In contrast to our setting, this information pertains to the cost of transferring surplus to the agent, rather than producing surplus. Furthermore, the private information pertains to the current period; information about the future is symmetrically held. This allows Li and Matouschek (2013) to apply recursive techniques. In contrast to the implicit downsizing costs in our setting, they find that every optimal equilibrium has the property of being sequentially optimal as well. The literature on implicit contracts also explores the
mal behavior of firms in the case of asymmetric information on the marginal profitability of employment (see Hart (1983), Azariadis (1983), or Grossman and Hart (1983)). There as well, inefficiently low employment in bad states of the world serves as a commitment device not to under-report the state of the world. This, however, is the consequence of an optimal risk-sharing arrangement between a risk-averse firm and its risk-averse workers.
The rest of the paper is set up as follows: Section 2 introduces the model; Section 3 reviews some benchmarks, in particular the case of public informa-tion; Section 4 presents the main results; Section 5 analyzes the impact of different hypotheses concerning timing, while Section 6 discusses an extension to persistent shocks. Section 7 concludes. Proofs not given in the text can be found in the Appendix.
2
The Model
Players. There is one principal (“she”) and one agent (“he”), who are both risk neutral and who interact repeatedly in periods t = 1, 2, · · · .
Actions. At the beginning of every period t, the principal makes an employment offer to the agent, consisting of a contractible wage wt∈ [− ¯w, ¯w],
where ¯w > 0 is assumed to be large enough. The agent then accepts (dt =
1) or rejects (dt = 0) the employment offer. If he accepts, the wage wt is
immediately paid. (If wt < 0, the agent pays −wt to the principal.) He
subsequently chooses his effort level nt ∈ R+. At the end of the period, the
principal can pay the agent a non-contractible, non-negative, bonus bt∈ [0, ¯b],
where ¯b > 0 is assumed to be large enough. Furthermore, she can send a non-verifiable cheap-talk message ˆθt∈ {θl, θh} at this time.5
Information. The public events (i.e. those that can be observed by both the principal and the agent) in period t are given by ht =
wt, dt, yt, bt, ˆθt
, where yt = g(nt). The production function g : R+ → R+ is C2, satisfies
g′ > 0 > g′′and lim
n↓0g′(n) = ∞, limn→∞g′(n) = 0. It is commonly known by
the players. A public history of length t−1, ht−1 (for t ≥ 2) collects the public
events up to, and including, time t − 1, i.e. ht−1 := (h
τ)t−1τ=1. We denote the set
of public histories of length t − 1 by Ht−1. (We set H0 = {∅}.) In each period,
a strategy for the agent specifies what wage offers to accept as a function
5Given our equilibrium concept (PPE in pure strategies, see below for details), the
of the previous public history, and what level of effort to exert if he accepts employment as a function of the previous public history and current-period wages. Formally, it is a sequence of mappingsσA
t
∞
t=1, where, for each t ∈ N,
σA
t = (dt, nt), and dt : Ht−1× [− ¯w, ¯w] → {0, 1}, (ht−1, wt) 7→ dt(ht−1, wt) and
nt: Ht−1× [− ¯w, ¯w] × {0, 1} → R+, (ht−1, wt, dt) 7→ nt(ht−1, wt, dt).
The principal additionally knows her type in period t + 1, θt+1 ∈ {θl, θh},
before deciding on the bonus payment bt in period t; the agent never learns
the realizations of the principal’s types. The values satisfy θh > θl > 0 and
are commonly known. We write θt= {θ
τ}tτ=1 for the sequence of realizations
of the principal’s types up to, and including, period t. The principal events in period t are given by ht =
wt, dt, yt, θt+1, bt, ˆθt
; that is, the principal learns about her period-t + 1 type already in period t, before paying the bonus in the respective period. A principal history of length t − 1, ht−1 (for t ≥ 2) collects
the principal events up to, and including, time t − 1, i.e. ht−1 := (h
τ)t−1τ=1. We
denote the set of principal-histories of length t − 1 by Ht−1. We assume that
the principal’s type in period t = 1 is commonly known to be θ1 = θh and thus
set H0 = {θh}. In each period, a pure strategy for the principal specifies his
wage offers as a function of the previous principal history, as well as his bonus payment and report as a function of the previous history, current-period wages and output, as well as his type in the next period. Formally, it is a sequence of mappings σP
t
∞
t=1, where, for each t ∈ N, σ P
t = (wt, bt, ˆθt), and wt : Ht−1 →
[− ¯w, ¯w], ht−1 7→ w
t(ht−1), bt : Ht−1 × [− ¯w, ¯w] × {0, 1} × R+ × {θl, θh} →
[0, ¯b], (ht−1, w
t, dt, yt, θt+1) 7→ bt(ht−1, wt, dt, yt, θt+1), with the restriction that
dt = 0 ⇒ bt(ht−1, wt, dt, yt, θt+1) = 0, and ˆθt: Ht−1× [− ¯w, ¯w] × {0, 1} × R+×
{θl, θh} → {θl, θh}, (ht−1, w
t, dt, yt, θt+1) 7→ ˆθt(ht−1, wt, dt, yt, θt+1). A pure
public strategy by the principal is a pure strategy which does not condition on her past private information, which is no longer payoff-relevant. Formally, a strategy σP
t
∞
t=1 is said to be a public strategy if, for each period t ∈ N,
it can be written σP t = ( ˜wt, ˜bt,˜ˆθt), where ˜wt : Ht−1 ×θl, θh → [− ¯w, ¯w], (ht−1, θ t) 7→ ˜wt(ht−1, θt), ˜bt : Ht−1 × [− ¯w, ¯w] × {0, 1} × R+× {θl, θh} → [0, ¯b] and ˜ˆθt: Ht−1 × [− ¯w, ¯w] × {0, 1} × R+× {θl, θh} → {θl, θh}.
While θ1 = θh, the principal’s types {θt}∞t=2 are i.i.d. across periods
(ex-cept in Section 6); for all t = 2, 3, · · · , the probability that θt= θh is q ∈ (0, 1).
The probability q, as well as the principal’s type in the first period, are com-mon knowledge.
Payoffs. If dt = 1, the principal’s period payoff in period t is given by
θtyt− wt− bt; the agent’s is given by wt− ntc + bt, where c > 0 is his marginal
cost of effort. If dt = 0, principal and agent get their outside option payoffs
in period t, which are set to zero. Both players discount future payoffs with the discount factor δ ∈ (0, 1).
Our solution concept is perfect Bayesian equilibrium in (pure) public strategies (PPE), as defined above. There are no long-term contracts or other means for the principal or the agent to commit to a certain course of action. In particular, the output yt is assumed to be non-verifiable.
Our objective is to find a PPE that maximizes the principal’s ex ante ex-pected profit Π1 among all PPE. As expected surplus can be transferred freely
through w1, the fixed wage in the first period, any equilibrium maximizing Π1
also maximizes the players’ joint surplus given the constraints, as shown by the following proposition, which parallels Levin’s (2003) Theorem 1.
Proposition 1 Suppose there exists a PPE leading to a joint surplus of s ≥ 0. Then, there exists a PPE giving the principal an expected payoff of π and the agent an expected payoff of u, for any (π, u) ∈ {(x, y) ∈ R+ : x + y = s}.
Proof. The proof follows that of Theorem 1 in Levin (2003) and is
therefore omitted.
As on-path equilibrium actions are completely determined by past type realizations, we shall replace histories as defined above with the history of previously reported types, which, on the equilibrium path, coincide with the history of past type realizations. We shall focus on truth-telling equilibria; i.e., on the equilibrium path, reported types will coincide with the history of past type realizations, θt = {θ
τ} t
τ=1. By our choice of equilibrium concept,
this is without loss in our main model of Sections 3 3-4. In a slight abuse of notation, we will thus write w(θt) for w
t(ht−1), and n(θt) for nt(ht−1, w(θt), 1),
the agent’s effort choice on the equilibrium path in period t given history θt.
In addition, we shall use superscripts h or l to indicate the type in period t + 1, given history θt, writing, for instance, bh(θt) for b
t(ht−1, w(θt), 1, yt, θh),
the principal’s on-path bonus payment after history θt, given that θ
t+1 = θh.
By the same token, we write Π(θt) = Πi(θt−1) for the principal’s expected
on-path profit, and U (θt) = Ui(θt−1) for the agent’s expected on-path utility,
at the beginning of period t, given the history of type realizations θtand given
Thus, we can write
Π(θt) =d(θt)θtg(n(θt)) − w(θt)
+ q −bh(θt) + δΠh(θt) + (1 − q) −bl(θt) + δΠl(θt) .
for the principal’s expected on-path profits for a given history of types θt, and
U (θt) =d(θt)w(θt) − n(θt)c + q bh(θt) + δUh(θt) + (1 − q) bl(θt) + δUl(θt) .
for the agent’s expected on-path utility in period t.
The following figure summarizes the timing within each period:
P makes offer A choosesnt θtg(nt) consumed by P θt+1 observed by P ˆ θt announced and bt paid to A
3
Some Benchmarks
In this section, we analyze a few natural benchmarks against which to measure our equilibrium.
First, suppose the principal and the agent acted cooperatively so as to maximize their joint surplus. Our assumptions on the production function g immediately imply that, in all periods t, the effort chosen would be equal to nF B(θ
t), with nF B(θt) being defined by the first-order condition
θtg′(nF B(θt)) = c.
For the remainder of the paper, we define nF B
h ≡ nF B(θh) and nF Bl ≡ nF B(θl).
Now, suppose that the agent’s effort choice was not just observable but also verifiable, while the principal’s type was her private information and both the principal and the agent maximized their own respective payoffs. Since the agent’s effort is verifiable, the principal and the agent can write a binding contract specifying, in each period t and given any history θt, that w
as well as bt = ntc if nt = nF B(θt) and bt = 0 otherwise. This sequence of
contracts implements first-best effort levels, and, since the principal collects the entire surplus, there is no sequence of contracts generating higher profits. In particular, as truth-telling gives her first-best profits, the principal has no incentive to lie.
If the game is played only once, the principal will never pay a positive bonus, whatever the agent’s effort level may have been. Anticipating this, the agent chooses n1 = 0, implying y1 = 0. In any equilibrium of the repeated
game, either party can always guarantee itself this static SPE payoff, which constitutes its minmax-payoff. As we are interested in the best possible PPE for the principal, it is without loss for us to focus on equilibria in which any observable deviation triggers this harshest possible punishment.6
3.1
Public Types
In this section, we suppose that the principal’s type is public information, while the agent’s effort is non-contractible. Thus, we assume that the agent observes next period’s type at the same time as the principal does, imply-ing that we here allow the agent to condition his strategy on the principal-histories rather than only the coarser public principal-histories. In this case, there is no informational asymmetry; agency problems arise merely on account of the non-contractibility of effort.
The agent always has the option of rejecting the principal’s offers forever, guaranteeing him a utility of 0. Therefore, after any history in any equilibrium, his expected utility will be at least 0, i.e., the following Individual Rationality constraint must hold, for all histories θt:
U (θt) ≥ 0. (IR)
Furthermore, after pocketing the fixed wages w(θt), the agent must find
it optimal to exert the level of effort he is supposed to exert in equilibrium, namely n(θt). Thus, his utility when exerting n(θt) must be at least as high
as his utility from exerting any other level of effort. As effort levels are ob-servable, it is without loss for us to focus on equilibria in which any deviation by the agent is punished in the harshest possible way, by giving him a
uation utility of 0; in such an equilibrium, therefore, any possible deviation is dominated by a deviation to an effort level of 0. Thus, the agent’s Incentive Compatibility Constraint is given by
−n(θt)c + q bh(θt) + δUh(θt) + (1 − q) bl(θt) + δUl(θt) ≥ 0. (IC)
It must also be optimal, after any history θt, for the principal to make the
bonus payments she is supposed to make in equilibrium. Indeed, as effort levels and bonus payments are not contractible, these must be self-enforcing. Again, we can focus without loss of generality on equilibria in which the principal is punished with a continuation profit of 0 whenever she does not pay out the bonus she is supposed to pay out; her best deviation in this case is to paying a bonus of 0. This yields the following dynamic enforcement constraints
−bh(θt) + δΠh(θt) ≥ 0 (DEh)
−bl(θt) + δΠl(θt) ≥ 0. (DEl)
It is standard to verify that (DEh) and (DEl) can equivalently be combined into a single constraint,
− qbh
(θt) + (1 − q)bl(θt) + δ qΠh
(θt) + (1 − q)Πl(θt) ≥ 0. (DE) The (DE) constraint states that the future benefits of honoring the relational contract must be sufficiently large for the principal that she is willing to bear today’s costs. Whereas these costs manifest themselves in (expected) bonus payments, the benefits are provided by the discounted difference between on-and off-path future profits. Since off-path profits, i.e., profits after a deviation, are zero, the benefits are identical to expected future profits.
Finally, it must be optimal for the principal to offer the equilibrium con-tract to the agent, i.e., Π(θt) ≥ 0. This, however, is already implied by the
(DE) constraint and our assumption that bonus payments are positive. Thus, our problem is to maximize Πh(∅), subject to (IR), (IC), and (DE),
through our choice of effort levels n(θt), wage and bonus payments w(θt), bl(θt)
and bh(θt), for all histories θt. The following lemma details some
characteris-tics of an optimal solution.
exists a profit-maximizing equilibrium in which the agent never gets a rent, that is,
• qbh(θt) + (1 − q)bl(θt) = n(θt)c and
• w(θt) = 0 for every history θt.
Furthermore, equilibrium effort only depends on the current state, that is, n(θt) = n(θ
t).
The lemma shows that there exists an optimal equilibrium in which the agent does not get a rent and the (IC) constraint will bind after any history. It also shows that the equilibrium is stationary. Hence, we can write n(θh) and
n(θl) for the respective equilibrium effort levels in any period t. The reason
for this is that, in the case of observable types, every deviation is observable; there is therefore no reason to burn any surplus on the equilibrium path of play.
Note that, as is also the case e.g. in Levin (2003) or MacLeod and Mal-comson (1989), enforceable effort in any given period does not depend on the current type but only on expected future profits. Indeed, current output is already sunk when the principal decides on the bonus payment. Optimal ef-fort, on the other hand, depends on today’s type. This tension delivers the intuition for the following proposition, which summarizes a profit-maximizing equilibrium with public types.
Proposition 2 Assume the firm’s type is publicly observable. Then, there are levels of the discount factor, δ and δ, with 0 < δ < δ < 1, such that
• n(θh) = nF B h and n(θl) = nF Bl for δ ≥ δ; • n(θl) = nF B l < n(θh) < nF Bh for δ < δ < δ; • n(θh) = n(θl) ≤ nF B l for δ ≤ δ.
If δ is high enough, the first best is achievable. For intermediate levels of the discount factor, nF B
h is no longer enforceable, while nF Bl still is. In
this case, the highest enforceable effort level is chosen in all periods t in which θt = θh, while nF Bl is enforced in all periods τ in which θτ = θl.
If the discount factor is so low that even nF B
l can no longer be enforced,
principal’s credibility today depends on next period’s type. Thus, she can credibly commit to a higher bonus payment if tomorrow’s type is high. If (DE) binds, it is indeed (strictly) optimal to have bh(θt) > bl(θt) because of
the agent’s risk neutrality.
4
Private Types
Now, let us assume that the principal’s type is her private information. Thus, she has to be given incentives not to misrepresent her true type. A straight-forward response would be to make the bonus payment independent of next period’s type; however, while feasible, such an approach is generally not opti-mal. In the following, we will explore how asymmetric information on future profits affects the properties of a profit-maximizing relational contract.
In truth-telling equilibrium, the principal needs sufficient incentives to reveal her type in every period. Specifically, after any history θt, it must be
optimal for her to pay out bh(θt) (rather than bl(θt)) if tomorrow’s state is
high, and bl(θt) (rather than bh(θt)) if tomorrow’s state is low; other bonus
payments never occur on the path of play and can therefore be deterred by threatening the principal with a continuation profit of 0. Lest punishment be triggered, once the principal has paid out bl(θt) at the end of period t, she can
only induce effort nl(θt) in period t + 1.7
Because, for any strategy choice by the agent, the principal always has a best response which is a public strategy, we only need to check the princi-pal’s incentives to deviate to another public strategy. Furthermore, thanks to discounting, the One-Deviation principle applies in our setting (see Hendon, Jacobsen, and Sloth (1996)). Therefore, if tomorrow’s state is high but the principal pays out the low-type bonus (or reports ˆθt+1 = θl, in case they are
7Note that a formal mechanism to transmit messages would not be required, whenever
the size of the bonus depends on tomorrow’s type, i.e. bh
(θt
) 6= bl
(θt
). In this case, bonus payments serve as a message and also determine next period’s equilibrium effort. In our equilibrium, whenever the principal’s report in period t + 1 does not correspond to the bonus having been paid in period t, punishment is triggered. When bh
(θt ) = bl (θt ) while nh (θt ) 6= nl (θt
), a message is needed to tell the agent which level of effort to choose in period t + 1.
equal) instead, her continuation payoff in period t + 1 can be written as ˜
Πl(θt) =θhg(nl(θt)) − wl(θt)
+ q −blh(θt) + δΠlh(θt) + (1 − q) −bll(θt) + δΠll(θt) ,
where the second superscript describes the type in period t + 2.
By the same token, if tomorrow’s state is low but the principal pays out the high-type bonus instead, her continuation payoff in period t + 1 is
˜
Πh(θt) =θlg(nh(θt)) − wh(θt)
+ q −bhh(θt) + δΠhh(θt) + (1 − q) −bhl(θt) + δΠhl(θt) .
Therefore, the principal is willing to tell the truth in equilibrium following history θt if and only if
−bh
(θt) + δΠh(θt) ≥ −bl(θt) + δ ˜Πl(θt) (TTh) −bl(θt) + δΠl(θt) ≥ −bh(θt) + δ ˜Πh(θt). (TTl)
As ˜Πl(θt) = Πl(θt)+θhg(nl(θt))−θlg(nl(θt)) and ˜Πh(θt) = Πh(θt)−θhg(nh(θt))+
θlg(nh(θt)), we can rewrite these constraints as follows:
−bh(θt) + δΠh(θt) ≥ −bl(θt) + δΠl(θt) + δg(nl(θt)) θh − θl
(TTh) −bl
(θt) + δΠl(θt) ≥ −bh(θt) + δΠh(θt) − δg(nh(θt)) θh− θl .
(TTl) Thus, the principal’s objective is to maximize Π(θ1) = θhg(n(θ1)) −
w(θ1) + q −bh(θ1) + δΠh(θ1) + (1 − q) −bl(θ1) + δΠl(θ1), where θ1 = θ 1 =
θh, subject to (DEh), (DEl), (TTh), (TTl), (IR) and (IC) at each history θt.
As we show in Appendix B.1, this optimization problem can be substan-tially simplified. First, the (DEh) constraint can be omitted because it is al-ways more tempting for the principal to underreport tomorrow’s type than to shut down. Furthermore, the agent never gets a rent, and the (IC) constraint always holds as an equality. Moreover, bh(θt) ≥ bl(θt), which implies that the
principal will never want to claim that the agent’s productivity tomorrow is higher than it actually is; i.e., the (TTl) constraint can be omitted. Thus, on the principal’s side, we are left with only the (DEl) and (TTh) constraints.
We further show in the Appendix that these constraints can equivalently be combined into one, and that consequently nh(θt) will be independent of θt,
while nl(θt) will only depend on the number i ∈ {0, 1, 2, ...} of consecutive
low shocks after the last high period. Therefore, we write nl
i to describe
low-type effort levels. Thus, the optimization problem boils down to choosing nh, nl i i∈N so as to maximize Πh = 1 − δ(1 − q) 1 − δ θ h g(nh) − nhc+1 − δ(1 − q) 1 − δ δ(1−q) ∞ X i=0 (δ(1 − q))i θlg(nli) − n l ic , subject to −nh c + δ qΠh+ (1 − q)Πl0 ≥ δqg(nl 0) θ h− θl . (ECh) and −nl ic + δ qΠh + (1 − q)Πli+1 ≥ δqg(nli+1) θh− θl (ECli) for all i ∈ N.
As mentioned above, the (EC) constraints are obtained by combining (DEl) and (TTh) constraints for the respective effort levels. The left-hand side of an (EC)-constraint is identical to the left-hand side of the (DE)-constraint with public types. It weighs the cost of compensating the agent for his effort costs against discounted expected future profits. With public types, this left-hand side had to exceed 0 for the principal to be willing to incur the cost of compensating the agent for his effort costs. With private types, by contrast, this has to be weakly greater than δqg(nl) θh− θl ≥ 0, which is an expression
for the principal’s information rent. Indeed, if (DE) constraints bind, the principal would like to transfer her entire future profits to the agent. But this is not feasible if the principal’s type tomorrow is θh (which happens with
probability q), because she always has the option of falsely claiming that the type is θl. If she does so, she will get θhg(nl) in the next period, rather than
just θlg(nl), which determines the bonus the principal is supposed to pay. As
(EC) shows, it is on account of this information rent that a given level of effort is harder to implement with private types.
(EC) constraints also imply that optimal efforts are the same in all high periods. The reason is that there is no trade-off with respect to effort levels in
high periods. Choosing them closer to the first-best benchmark both increases the objective and relaxes the constraint; indeed, making a high period more attractive makes the principal less inclined falsely to claim to be in a low period. The effort level in a low period, by contrast, depends on the history, albeit only via the distance of the current period to the last previous high period. The reason is that there is a trade-off with respect to the effort level in a low period. Making a low period less attractive lowers the objective but relaxes the constraint as it makes it less enticing for the principal falsely to claim to be in a low period. Thus, the optimal effort level in a given low period depends on the optimal effort level in the previous period.
In conclusion, the agency problem here consists not only in the non-verifiability of the agent’s performance measures, but also in the necessity of preventing the principal from claiming her type to be lower than it actually is in order to save on her bonus payments. Lying generally does not come for free, though, because only the respective low-type effort can be implemented in the subsequent period. Thus, for the same reason as in the case of public types, it can still be optimal to have bh(θt) > bl(θt), despite the principal’s temptation
to lie. Then, the principal’s tradeoff boils down to a comparison of today’s benefits of a deviation (a lower bonus payment) with tomorrow’s costs (a lower output). This aspects adds another dimension to the credibility problem typical for relational contracts, in a sense that the principal’s credibility is reduced by the information rent she can always secure herself because of her private information. As we shall see below, tweaking tomorrow’s costs of lying, by adjusting the output level given tomorrow’s type is low, can be a way of boosting the principal’s credibility today.
Our first result shows that if the discount factor is close enough to 1, the first best can be achieved.
Proposition 3 There exists a δ ∈ (0, 1) such that for all δ ≥ δ, the unique optimal equilibrium implies first-best effort levels nF B
h /nF Bl .
To get an intuition for the forces at play, recall that the (EC)-constraints in fact capture two distinct effects. On the one hand, there is the classical effect coming from the dynamic-enforcement constraints that the principal would never be willing to make a bonus payment exceeding the discounted expected value of the continuation of the relationship to her. As we have seen above, this constraint can only ever bind in our setting if the principal observes
the next period to be low. On the other hand, there is the need to incentivize the principal to tell the truth because a higher enforceable bonus when the next period is high may tempt the principal to lie in order to reduce her bonus payments in the current period. A straightforward response to this temptation is a reduction of bh(θt), accompanied by an appropriate increase of bl(θt) to
leave incentives for the agent unaffected. This, however, is restricted by δΠl,
which is the most the principal would be willing to pay given that tomorrow’s type is low. Yet, as δ, and hence δΠl, increase, it becomes possible to increase
bl without violating (DEl); this in turn reduces the principal’s incentives to
lie. The proposition now shows that, when δ is close enough to 1, the (EC) constraint will hold, and hence the principal will not have any incentives to lie or to renege on her bonus payment.
Our next proposition presents the first main result of this paper. It characterizes an optimal outcome, given that the discount factor is too low to implement nF B
h but high enough to implement n F B l .
Proposition 4 There exist discount factors δ and δ, with 0 < δ < δ < 1, such that, in an optimal equilibrium, for δ ∈ (δ, δ), nh and nl
0 are inefficiently
low: nl
0 < nF Bl < n
h < nF B
h , and, for all i ≥ 1, n l
i = nF Bl .
Note that, for the first-best solution, the (ECh) and (ECli) constraints are identical but for the first term, which is nF B
h and nF Bl , respectively. Thus,
as δ decreases, (ECh) starts binding before the (ECli) constraints do. When this happens, nF B
h is no longer implementable and n
h is hence reduced below
first-best levels. Yet, as Proposition 4 shows, nl
0 is reduced below nF Bl as well,
even though (ECl0) does not bind. This “overshooting” relaxes (ECh) and thus allows for a smaller reduction in nh than would otherwise be necessary.
Because the principal needs to be dissuaded from claiming that next period’s type is low when it is in fact high, low periods need to be rendered less attractive, and, in particular, those low periods that follow periods in which the principal needs a lot of credibility, i.e., high periods. A natural, surplus-neutral, way of achieving this goal would be to force the principal to make a transfer to the agent if he claims next period’s type to be low. However, such a transfer would relax (TTh), but tighten (DEl) to the same extent. Therefore, (EC) constraints, which are combinations of the respective (TTh) and (DEl) constraints, would not be relaxed.
To develop an intuition for this result, it is helpful to split up (ECh) again and to take a look at its individual components, the (TTh) and (DEl) constraints for nh: −bh(nh) + δΠh ≥ −bl(nh) + δΠl 0+ δg(nl0) θh− θl (TTh) −bl (nh) + δΠl0 ≥ 0 (DEl) Consider an arbitrary effort level nh < nh
F B together with bonuses bh(nh)
and bl(nh) such that (TTh) and (DEl) hold as equalities, i.e., (ECh) binds.
(Indeed, if only one of them was binding, for example (TTh), a first response would be to reduce bh(nh) by ε > 0 and to increase bl(nh) by q
(1−q)ε, which
would allow for a further increase in nh..) In order to relax (TTh), one could
pay a rent R to the agent following an announcement of a low state at the end of the period, an arrangement equivalent to making such a payment at the beginning of the next period. This reduces the right-hand side of (TTh) by R, thereby relaxing (TTh) and allowing the principal to increase bh(nh) by
R as well. However, the principal also needs an incentive to pay R. Hence, the (DEl) constraint becomes −bl(nh) − R + δΠl
0 ≥ 0. As (DEl) was binding
before, bl(nh) must be reduced by R in order to keep (DEl) satisfied. But this
once again increases the right-hand side of (TTh) by R, making it necessary to reduce bh(nh) by the same amount (and hence to its original level) – and,
at the end, nothing has been gained.
Thus, (ECh), the combination of (TTh) and (DEl) constraints, can only be relaxed by downsizing costs if those hit a lying off-path principal harder than a principal who truthfully claims next period’s type to be low. Mere transfers cannot achieve this goal as we have just seen. However, the distor-tion of effort levels as proposed by Proposidistor-tion 4, which can be interpreted as implicit downsizing costs, hits a lying off-path principal harder than a truthful principal and therefore relaxes (ECh). To see that, assume that in the situ-ation considered in the previous paragraph, effort after an announcement of a low state is reduced by a small ε > 0 in the following period. This reduces Πl
0 – and consequently bl(nh) – by ε θlg′− c. However, the right-hand side
of (TTh) is decreased by εδg′ θh− θl, which allows for an increase in bh(nh)
by the same amount. Asθlg′(nl
F B) − c = 0, the resulting surplus destruction,
as well as the necessary reduction in bl(nh), are only of second order at nl F B.
even-tually relaxed. Therefore, it is optimal to use a reduction of nl
0 in order to
implement a larger nh. Thus, the game exhibits memory, and the equilibrium
is not sequentially optimal, in that nF B
l > nl0 would be implemented if the
game newly started with a low state.
This contrasts with the finding in Li and Matouschek (2013), where every optimal equilibrium is sequentially optimal. In our iid model, this distortion in effort levels only lasts a single period, and nl
i = nF Bl for i ≥ 1. This is
due to two reasons. First, reducing nl
i for i ≥ 1 would not allow for a further
increase in nh because the resulting distortions in later periods would hit
on-path and off-on-path principals alike.8 Second, for discount factors above δ, (ECl) constraints do not bind and first-best effort levels are feasible. Thus, implicit downsizing costs indeed optimally arise on the equilibrium path.
Given δ is below δ, ECli constraints also bind for i ≥ 1. This consid-erably complicates our maximization problem because all (ECli) constraints potentially interact: A higher nl
i+1 tightens (ECli), whereas a higher nli might
require a reduction of nl
i+1 and consequently relax (ECli+1). Therefore, we
have to consider infinitely many constraints. In the following, we derive a number of properties of effort levels nl
i if δ < δ. Due to the complexity of the
problem, we restrict ourselves to the case qθh ≥ θl:
Proposition 5 Assume qθh ≥ θl. There exists a left-neighborhood of δ such
that optimal effort levels nl
i < nlF B, i ≥ 1, are characterized by one of the
following cases: • nl
j = nl1 for all odd j and nlι = nl2 for all even ι, with nl1 > nl2;
• nl
1 > nl3 > nl5 > ... and nl2 < nl4 < nl6 < ..., with supj∈Nnl2j ≤
infj∈Nnl2j−1;
• qθh = θl⇔ nl
i = nli+1 for all i ≥ 1.
We prove this proposition by Lemmata 10 - 23 in the Appendix. It shows that, unless qθh = θl, effort levels oscillate, with either a constant or
a decreasing amplitude, starting at their highest level nl
1. If qθh = θl, by
contrast, effort levels nl
i (i ≥ 1) remain constant, as for intermediate discount
factors. We still observe overshooting in this region, as nl
0 is constrained only
by ECh, while ECl0 is slack. Furthermore, nh < nh
F B and nl0 < nl1.
8As we shall see in Section 6, distortions last longer when types are (fully or partially)
5
Timing
Here, we vary the timing of the revelation of next period’s type. First, we show that a later revelation – the type of period t is revealed at the beginning of period t – increases the principal’s profits compared to our main case. Then, private information is not costly and the outcome equivalent to the case of public information. Second, we assume that the principal observes the type of period t + 1 already at the beginning of period t, before the agent exerts period-t effort. In this case, private information is costly, but the nature of the costs and the principal’s response substantially differs. Moreover, our overshooting result in the previous section relies on there being no possibility of monetary transfers in between the time of the agent’s effort choice and the revelation of private information to the principal.
5.1
θ
tRevealed at Beginning of Period
t
Assume that the type of period t is revealed at the beginning of period t (this is equivalent to having θt+1 revealed in period t, but after bt has been
paid). First, we derive a profit-maximizing equilibrium for the case of public information. Then, we show that the associated effort and compensation levels also satisfy the truth-telling constraints under private information.
In contrast to before, the bonus btis not a function of next period’s type
anymore and hence is certain (on the equilibrium path) at the time of effort choice. In period t, the agent’s future compensation might still depend on θt+1,
though, through the fixed wage wt+1. But it turns out that it remains (weakly)
optimal to use only certain period-t bonus payments to reward period-t effort. Here, we consider a quasi-stationary equilibrium in the sense that bonus and effort are only a function of today’s type. The wage might be a function of today’s and yesterday’s type, if it is used to provide incentives for yesterday’s effort. We use left and right superscripts to describe wages (and profits) as functions of θt−1 (left) and θt (right). For example, if the type in both periods
h Πh = θhg(nh) − bh− h wh+ δΠh lΠh = θhg(nh) − bh− lwh+ δΠh h Πl = θlg(nl) − bl− h wl+ δΠl l Πl = θlg(nl) − bl− lwl+ δΠl,
with Πh = qhΠh+ (1 − q)hΠland Πl= qlΠh+ (1 − q)lΠl. The agent’s utilities
are described accordingly.
We maximize Πh, subject to the following constraints:9
−nhc + bh+ δUh ≥ 0 (ICh)
−nlc + bl+ δUl ≥ 0 (ICl)
−bh+ δΠh ≥ 0 (DEh) −bl+ δΠl ≥ 0. (DEl)
First, we show that it is weakly optimal only to use the bonus to provide incentives, while setting wages equal to zero: If any fixed wages were strictly positive, a reduction accompanied by a corresponding increase of the respec-tive bonus would leave all constraints unaffected (for example, if hwh > 0,
reducing hwh by a small ε > 0 and increasing bh by δqε has no effect on ICh
and DEh) and not decrease profits. Furthermore, as in Lemma 1, we can show that it is feasible and optimal to set bh = nhc and bl = nlc. Then, the two
remaining constraints are
−nhc + δq θ hg(nh) − nhc + (1 − q) θlg(nl) − nlc (1 − δ) ≥ 0 −nlc + δq θ hg(nh) − nhc + (1 − q) θlg(nl) − nlc (1 − δ) ≥ 0,
which are the same as in our main setting with public information. Therefore, profit-maximizing effort levels are also characterized by Proposition 2, with
9
Note that maximizing any other of the above profit streams would yield identical out-comes because the equilibrium – as we will see below – is now sequentially efficient.
levels of the discount factor, δ and δ ( 0 < δ < δ < 1), such that nh = nF B h and
nl = nF B
l for δ ≥ δ; nl = nF Bl < nh < nF Bh for δ < δ < δ; and nh = nl ≤ nF Bl
for δ ≤ δ.
If the type is the principal’s private information, additional truth-telling constraints, now imposed at the beginning of a period, must hold:
θhg(nh) − bh− hwh+ δΠh ≥ θhg(nl) − bl− hwl+ δΠl (TThh) θhg(nh) − bh− l wh+ δΠh ≥ θh g(nl) − bl− l wl+ δΠl (TTlh) θlg(nl) − bl− hwl+ δΠl≥ θlg(nh) − bh− hwh+ δΠh (TThl) θlg(nl) − bl− l wl+ δΠl≥ θl g(nh) − bh− l wh+ δΠh (TTll) To show that these constraints can be omitted, we plug the results from the case with public information, hwh = hwl = lwh = lwl = 0 and bh = nhc
and bl = nlc, into the conditions. Then, Πh = Πl = q(θhg(nh)−nhc)+(1−q)(θlg(nl)−nlc) (1−δ) ,
and the constraints become
θhg(nh) − nhc ≥ θhg(nl) − nlc (TThh)
θhg(nh) − nhc ≥ θhg(nl) − nlc (TTlh)
θlg(nl) − nlc ≥ θlg(nh) − nhc (TThl) θlg(nl) − nlc ≥ θlg(nh) − nhc. (TTll)
These are naturally satisfied for the respective effort levels.
To understand the intuition behind this result, note that in our bench-mark case, it would also be feasible to make the agent’s compensation indepen-dent of the realization of next period’s type. But such a payment structure would leave some slackness in the dynamic enforcement constraints, which could be utilized in order to increase implementable effort. At some point, however, truth-telling constraints start to bind, leading to the structure of the profit-maximizing equilibrium that we have derived in Section 4. Here, by contrast, the agent’s compensation can be independent of next period’s type while fully exhausting dynamic enforcement constraints. Thus, imple-mentable effort cannot be further increased. Therefore, it is optimal to make the agent’s compensation (conditional on effort) independent of the realization of next period’s type.
5.2
θ
t+1Revealed at Beginning of Period
t
Now, we describe the properties of a profit-maximizing equilibrium for the case that the type of period t + 1 is already revealed at the beginning of period t, before effort nt is chosen by the agent. First, we analyze how this information
structure affects a profit-maximizing equilibrium under public information. 5.2.1 Public Information
If θt+1 is (publicly) revealed at the beginning of period t, nt will be a function
not only of today’s, but generally also of tomorrow’s, type. This is because enforceable effort in a given period is a function of expected future profits. A high type tomorrow is associated with higher expected future profits and thus a higher enforceable effort level today. By standard arguments, it is without loss to analyze otherwise stationary equilibria. We therefore use superscripts to indicate equilibrium values as functions of this and next period’s types. For example, nhh is equilibrium effort in case today’s and tomorrow’s type are
high, nhl is equilibrium effort if today’s type is high and tomorrow’s type is
low, and so on.
Then, on-path profit streams can take one of the four values
Πhh= θhg(nhh) − whh− bhh+ δΠh
Πhl = θhg(nhl) − whl− bhl+ δΠl
Πlh = θlg(nlh) − wlh− blh+ δΠh Πll = θlg(nll) − wll− bll+ δΠl,
where Πh ≡ qΠhh + (1 − q)Πhl and Πl ≡ qΠlh + (1 − q)Πll. The agent’s
utilities are defined equivalently. Bonus payments are bounded by dynamic enforcement constraints,
−bhh + δΠh ≥ 0 (DEhh) −bhl+ δΠl ≥ 0 (DEhl) −blh + δΠh ≥ 0 (DElh) −bll+ δΠl≥ 0, (DEll) whereas effort levels are bounded by incentive compatibility constraints, −nhhc + bhh+ δUh ≥ 0 (IChh) −nhl c + bhl+ δUl ≥ 0 (IChl) −nlhc + blh+ δUh ≥ 0 (IClh) −nll c + bll+ δUl ≥ 0. (ICll) Now, although the bonus is a function of next period’s type, it is certain at the time of the agent’s effort choice. This is different from the main part of our paper, where next period’s type is revealed immediately before today’s bonus is paid, and therefore uncertain at the time of the agent’s effort choice. Furthermore, for reasons similar to above (Lemma 1), it is feasible and weakly optimal to set whh = whl = wlh = wll = 0 and let (IC) constraints hold as
equalities. Therefore, bhh= nhhc, bhl = nhlc, blh = nlhc and bll = nllc.
Again, our objective is to maximize Πh, now subject to
−nhhc + δΠh ≥ 0 (DEhh)
−nhl
c + δΠl≥ 0 (DEhl) −nlhc + δΠh ≥ 0 (DElh) −nllc + δΠl ≥ 0. (DEll)
It is immediate that Πh ≥ Πl, i.e. that a high type is associated with higher profits. Therefore, θt+1 = θh allows for a credible promise of a higher
bonus, and therefore for the implementation of a higher effort level, in period t. The desired effort levels if today’s type are high (nhhand nhl) are also larger
the discount factor is sufficiently close to 1, none of the constraints bind and first-best levels nhh = nhl = nF B
h and nll = nlh = nF Bl can be implemented.
For a lower discount factor, (DEhl) will eventually bind, and nhl < nhh= nF B h .
For even lower discount factors, (DEhh) and/or (DEll) will at some point bind as well, and so on. These considerations are summarized in Lemma 2.
Lemma 2 Assume θt+1is publicly revealed at the beginning of period t. Then,
there are levels of the discount factor, δ, ˜δ and δ, with 0 < δ < ˜δ < δ < 1, such that • nhh= nhl = nF B h > n ll = nlh = nF B l for δ ≥ δ; • nhl < nhh= nF B h and nll= nlh = nF Bl for ˜δ ≤ δ < δ.
• For δ ≤ δ < ˜δ, there are levels of the discount factor δh and δl, such
that – nhl < nhh < nF B h for δ < δh; – nll < nlh = nF B l for δ < δl ; • nhl < nhh < nF B
h and nll < nlh < nF Bl for δ < δ; in this case, nll =
nhl < nlh = nhh.
The early revelation of information is costly compared to a later reve-lation – because no “cross-subsidization” of high future profits to low future profits is feasible anymore. If information is revealed later (like in the previous section), the resulting uncertainty allows us to use potential high future profits to motivate effort also in case future profits are actually low. Here, a binding (DEhl) constraint cannot be relaxed by a potential slackness of (DEhh), as would be the case if information was revealed later.
5.2.2 Private Information
If next period’s type is only privately revealed to the principal at the beginning of the present period, the relevant trade-off in truth-telling equilibrium is different from the main part of this paper. There, the principal is tempted to underreport her type because this results in a lower bonus payment to the agent in the present period, at the cost of a distorted production in the next period. The current effort level is unaffected by a lie of the principal,
as the corresponding output has already been realized. If next period’s type is revealed at the beginning of the present period, however, under-reporting tomorrow’s type already results in a lower output today. Therefore, a lie is associated with present and future production inefficiencies. The resulting costs make the principal’s temptation to under-report her type vanish, and overshooting as a consequence of downsizing is not needed to induce truth-telling. Truth-telling constraints can still severely constrain profits, though, due to an issue that was absent before: Because having a high type in the next period potentially allows for higher effort, and consequently higher profits, today, the principal might be tempted falsely to claim that tomorrow’s type is high – and then to renege on the promised payment. It turns out that this constraint in fact prevents the principal from achieving higher profits if tomorrow’s type is high. Profits will only be a function of today’s type, and will always be constrained by δΠl – no matter if next period’s type is actually high or low.
To formally derive this result, we keep the notation from our analysis with public types. Though this restriction is not without loss of generality here, we continue to focus on the same kind of quasi-stationary equilibria as with public types, where fixed wages equal zero and (IC) constraints bind. We will show below that, in contrast to before, the relevant truth-telling constraints can now either be satisfied by a reduction of effort levels, or by an ex-ante payment made to the agent. If these payments can be extracted by the principal at the beginning of the game, such an agreement would indeed maximize the principal’s profits.
Now, two types of truth-telling constraints arise. First, the principal might misreport her type and then proceed with play as prescribed by equi-librium (like in our main case). This yields the constraints
Πhh ≥ ˜Πhl (TThh) Πhl ≥ ˜Πhh (TThl)
Πlh ≥ ˜Πll (TTlh) Πll ≥ ˜Πlh, (TTll)
where ˜
Πhh = θhg(nhh) − nhhc + δhqθlg(nhh) − nhhc + δΠh+ (1 − q)θlg(nhl) − nhlc + δΠli = Πhh− δ θh− θl qg(nhh) + (1 − q)g(nhl)
are the principal’s profits in case today’s type is high and tomorrow’s type is low, but where she falsely reports tomorrow’s type to be high.
The respective values ˜Πhl, ˜Πlh and ˜Πll are obtained in similar fashion.
The second kind of truth-telling constraints prevent the principal from misre-porting her type and subsequently shutting down.
These constraints are
θhg(nhh) − nhhc + δΠh ≥ θh g(nhl) (TThh2) θhg(nhl) − nhlc + δΠl ≥ θhg(nhh) (TThl2) θlg(nlh) − nlhc + δΠh ≥ θl g(nll) (TTlh2) θlg(nll) − nllc + δΠl ≥ θlg(nlh) (TTll2)
Note that these kinds of constraints are not needed in our main case. There, next period’s type is revealed after today’s effort and output have been realized. They are thus sunk when the principal’s announces next pe-riod’s type. Therefore, these constraints coincide with the respective dynamic enforcement constraints.
Finally, (DE) constraints as specified in the previous section with public information must hold. This yields
Proposition 6 Assume θt+1 is privately revealed at the beginning of period
t. Then, among the class of equilibria in which the agent does not get a rent and faces binding (IC) constraints, Πh is maximized when nhh = nhl ≡ nh
and nlh = nll ≡ nl. Moreover, there exist discount factors δ and δ, with
0 < δ < δ < 1, such that • nh = nF B h and nl = nF Bl for δ ≥ δ; • nl= nF B l < n h < nF B h for δ < δ < δ; • nl= nh ≤ nF B l for δ ≤ δ in this equilibrium.
Importantly, effort is only a function of today’s type. High future profits cannot be used to implement higher effort today. If this were the case, the principal would have an incentive to misreport her type and then shut down. Moreover, the temptation now lies in over-reporting one’s type because this would be associated with higher productivity.
Note that for discount factors such that the first-best effort cannot be implemented, it only matters that the principal’s profits not be larger if to-morrow’s type is high. Instead of equalizing effort levels, we could also have nhh > nhl, together with a payment to the agent before his effort choice, in
the form of a positive fixed wage whh. Then, TThl2, which is the tighter
constraint, becomes
θhg(nhl) − nhlc + δΠl≥ −whh
+ θhg(nhh), (TThl2) which is satisfied for θhg(nhh) − whh = θhg(nhl) (given dynamic enforcement
constraints hold). Thus, in order to make use of higher future profits and induce the agent to work harder, the principal immediately has to pay him for the extra effort. If the principal is able to extract the expected value of these payments at the beginning of the game, this equilibrium generates higher expected profits than that of Proposition 6.
6
Persistent Shocks
So far, we have assumed that the principal’s types are iid across periods. In this section, we show that implicit downsizing costs may also obtain if shocks are persistent – within our initial setup where θt+1 is revealed to the principal
before the period-t bonus is paid, but after effort has been exerted. First, we explore permanent shocks. We assume that the principal starts out with a high type, and that the type remains high for another period with time-invariant probability q. With probability 1 − q, the type switches to low and remains low forever. Then, we argue that the case with shocks that are persistent but not permanent yields similar results.
6.1
Permanent Shocks
As the problem conditional on the type still being high is stationary, it is without loss for us to restrict attention to equilibria in which actions do not depend on calendar time. Therefore, equilibrium high-type effort is constant, whereas low-type effort depends on the distance in time to the (now perma-nent) switch from high to low. Thus, equilibrium profits can be written
Πh = θhg(nh) − qbh− (1 − q)bl
0− wh + δqΠh+ δ(1 − q)Πl0
Πl
i = θlg(nli) − bli+1− wil+ δΠli+1,
where wh and wl
i are defined analogously to nh and nli.
The objective is to maximize Πh, subject to the following constraints.
First, the dynamic enforcement (DE) constraints must be satisfied for bh and
all bl i: −bh + δΠh ≥ 0 (DEh) −bl i+ δΠ l i ≥ 0∀i ≥ 0. (DEli)
As long as the principal has not announced a switch to the low state, the following truth-telling constraints must hold in a truth-telling equilibrium:
−bh+ δΠh ≥ −bl 0+ δ ˜Πl0 (TTh) −bl 0+ δΠ l 0 ≥ −b h + δ ˜Πh. (TTl) After claiming that the state has switched to θl, the principal has to claim a
state of θl in all subsequent periods. This gives us
˜ Πl i = θ hg(nl i)−b l i+1−wil+δ h q ˜Πl i+1+ (1 − q)Πli+1 i = Πl i+ ∞ X τ=0 (δq)τg(nl τ)(θ h−θl).
Note that our formulation of ˜Πl
i takes into account that the principal does not
renege after falsely having announced a switch to state θl in the past. This
requires −bl
i+ δ ˜Πli ≥ 0, which holds given the (DEli) constraints and Πli < ˜Πli.
